3.442 \(\int \left (2-x^2\right ) \sqrt [4]{6 x-x^3} \, dx\)

Optimal. Leaf size=17 \[ \frac{4}{15} \left (6 x-x^3\right )^{5/4} \]

[Out]

(4*(6*x - x^3)^(5/4))/15

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Rubi [A]  time = 0.00850163, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048 \[ \frac{4}{15} \left (6 x-x^3\right )^{5/4} \]

Antiderivative was successfully verified.

[In]  Int[(2 - x^2)*(6*x - x^3)^(1/4),x]

[Out]

(4*(6*x - x^3)^(5/4))/15

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Rubi in Sympy [A]  time = 1.63635, size = 12, normalized size = 0.71 \[ \frac{4 \left (- x^{3} + 6 x\right )^{\frac{5}{4}}}{15} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((-x**2+2)*(-x**3+6*x)**(1/4),x)

[Out]

4*(-x**3 + 6*x)**(5/4)/15

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Mathematica [A]  time = 0.0167495, size = 16, normalized size = 0.94 \[ \frac{4}{15} \left (-x \left (x^2-6\right )\right )^{5/4} \]

Antiderivative was successfully verified.

[In]  Integrate[(2 - x^2)*(6*x - x^3)^(1/4),x]

[Out]

(4*(-(x*(-6 + x^2)))^(5/4))/15

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Maple [A]  time = 0.007, size = 20, normalized size = 1.2 \[ -{\frac{4\,x \left ({x}^{2}-6 \right ) }{15}\sqrt [4]{-{x}^{3}+6\,x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-x^2+2)*(-x^3+6*x)^(1/4),x)

[Out]

-4/15*(-x^3+6*x)^(1/4)*x*(x^2-6)

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Maxima [A]  time = 0.733525, size = 18, normalized size = 1.06 \[ \frac{4}{15} \,{\left (-x^{3} + 6 \, x\right )}^{\frac{5}{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(-x^3 + 6*x)^(1/4)*(x^2 - 2),x, algorithm="maxima")

[Out]

4/15*(-x^3 + 6*x)^(5/4)

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Fricas [A]  time = 0.259894, size = 27, normalized size = 1.59 \[ -\frac{4}{15} \,{\left (x^{3} - 6 \, x\right )}{\left (-x^{3} + 6 \, x\right )}^{\frac{1}{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(-x^3 + 6*x)^(1/4)*(x^2 - 2),x, algorithm="fricas")

[Out]

-4/15*(x^3 - 6*x)*(-x^3 + 6*x)^(1/4)

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Sympy [A]  time = 0.689727, size = 31, normalized size = 1.82 \[ - \frac{4 x^{3} \sqrt [4]{- x^{3} + 6 x}}{15} + \frac{8 x \sqrt [4]{- x^{3} + 6 x}}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-x**2+2)*(-x**3+6*x)**(1/4),x)

[Out]

-4*x**3*(-x**3 + 6*x)**(1/4)/15 + 8*x*(-x**3 + 6*x)**(1/4)/5

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GIAC/XCAS [A]  time = 0.274269, size = 18, normalized size = 1.06 \[ \frac{4}{15} \,{\left (-x^{3} + 6 \, x\right )}^{\frac{5}{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(-x^3 + 6*x)^(1/4)*(x^2 - 2),x, algorithm="giac")

[Out]

4/15*(-x^3 + 6*x)^(5/4)